Consider again the Hotelling model with exogenous prices and endogenous locations. As before, let the product space be the unit interval, [0, 1]. As in the problem set, let there be THREE firms, known as Firm 1, Firm 2, and Firm 3. All three firms are required to set the same price, p. for their product, and they all have zero production costs. The three firms simultaneously choose 'location'. That is, each of the firms chooses a location f; [0,1]. The consumers observe the set of locations, f = (fi, fo, f3), and then each consumer non-strategically purchases from the firm whose location closest to her. Assume that if two or more firms minimise this distance then a consumer is equally likely to purchase from each. The difference with this question, as compared to the problem set, is that Firms 1 and 2 are owned by the same subsidiary, who chooses the locations of Firms 1 and 2 jointly to maximise their profit. Firm 3 stands on its own, competing against the subsidiary. In all the following questions, you may either argue mathematically or with words, providing intuition. If you wish to use a result from the lecture slides, please state it at the outset. (a) Is there an equilibrium with all three firms at separate locations? (b) Is there an equilibrium with all three firms at the same location? () Is there an equilibrium with Firms 1 and 2 at the same location and Firm 3 at a different location? (d) Is there an equilibrium with one firm from the subsidiary at the same location as Firm 3 (e.g. Firm 1) and the other firm from the subsidiary at a different location (e.g. Firm 2)? (e) Comment on the existence (or lack thereof) of a pure strategy equilibrium in this model. Do you think there is a mixed strategy equilibrium